... except for the rules of arithmetic, a total being the sum of the
various categorized flight times.

Still just speculation on your part. There is no reason to think that
the listed categorized kinds of time are exhaustive.

The person the OP mentioned got valuable experience, but which may not
fall into any official category.

He can certainly log it as "Total Time" because Total Time can mean
whatever he wants it to mean.

Much of the confusion over this subject would go away if people would
use the proper terms.